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10 GEOMETRIC GRAPH THEORY J´Anos Pach

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10 GEOMETRIC GRAPH THEORY J´anos Pach INTRODUCTION In the traditional areas of graph theory (Ramsey theory, extremal graph theory, random graphs, etc.), graphs are regarded as abstract binary relations. The relevant methods are often incapable of providing satisfactory answers to questions arising in geometric applications. Geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edges (or, more generally, by edges represented by simple Jordan arcs). It is a fairly new discipline abounding in open problems, but it has already yielded some striking results that have proved instrumental in the solution of several basic problems in combinatorial and computational geometry (including the k-set problem and metric questions discussed in Sections 1.1 and 1.2, respectively, of this Handbook). This chapter is partitioned into extremal problems (Section 10.1), crossing numbers (Section 10.2), and generalizations (Section 10.3). 10.1 EXTREMAL PROBLEMS Tur´an’s classical theorem [Tur54] determines the maximum number of edges that an abstract graph with n vertices can have without containing, as a subgraph, a complete graph with k vertices. In the spirit of this result, one can raise the follow- ing general question. Given a class of so-called forbidden geometric subgraphs, what is the maximum number of edgesH that a geometric graph of n vertices can have without containing a geometric subgraph belonging to ? Similarly, Ramsey’s the- orem [Ram30] for abstract graphs has some natural analoguesH for geometric graphs. In this section we will be concerned mainly with problems of these two types. GLOSSARY Geometric graph: A graph drawn in the plane by (possibly crossing) straight- line segments; i.e., a pair (V (G), E(G)), where V (G) is a set of points (‘vertices’), no three of which are collinear, and E(G) is a set of segments (‘edges’) whose endpoints belong to V (G). Convex geometric graph: A geometric graph whose vertices are in convex po- sition; i.e., they form the vertex set of a convex polygon. Cyclic chromatic number of a convex geometric graph: The minimum number χc(G) of colors needed to color all vertices of G so that each color class consists of consecutive vertices along the boundary of the convex hull of the vertex set. Convex matching: A convex geometric graph consisting of disjoint edges, each of which belongs to the boundary of the convex hull of its vertex set. 257 Preliminary version (July 26, 2017). To appear in the Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, and C. D. Tóth (editors), 3rd edition, CRC Press, Boca Raton, FL, 2017. 258 J. Pach Parallel matching: A convex geometric graph consisting of disjoint edges, the convex hull of whose vertex set contains only two of the edges on its boundary. Complete geometric graph: A geometric graph G whose edge set consists of |V (G)| all 2 segments between its vertices. Complete bipartite geometric graph: A geometric graph G with V (G) = V V , whose edge set consists of all segments between V and V . 1 ∪ 2 1 2 Geometric subgraph of G: A geometric graph H, for which V (H) V (G) and E(H) E(G). ⊆ ⊆ Crossing: A common interior point of two edges of a geometric graph. (k,l)-Grid: k + l vertex-disjoint edges in a geometric graph such that each of the first k edges crosses all of the last l edges. It is called natural if the first k edges are pairwise disjoint segments and the last l edges are pairwise disjoint segments. Disjoint edges: Edges of a geometric graph that do not cross and do not even share an endpoint. Parallel edges: Edges of a geometric graph whose supporting lines are parallel or intersect at points not belonging to any of the edges (including their endpoints). x-Monotone curve: A continuous curve that intersects every vertical line in at most one point. Outerplanar graph: A (planar) graph that can be drawn in the plane with- out crossing so that all points representing its vertices lie on the outer face of the resulting subdivision of the plane. A maximal outerplanar graph is a triangulated cycle. Hamiltonian path: A path going through all elements of a finite set S. If the elements of S are colored by two colors, and no two adjacent elements of the path have the same color, then it is called an alternating path. Hamiltonian cycle: A cycle going through all elements of a finite set S. Caterpillar: A tree consisting of a path P and of some extra edges, each of which is adjacent to a vertex of P . CROSSING-FREE GEOMETRIC GRAPHS 1. Hanani-Tutte theorem: Any graph that can be drawn in the plane so that its edges are represented by simple Jordan arcs such that any two that do not share an endpoint properly cross an even number of times, is planar [Cho34, Tut70]. The analogous result also holds in the projective plane [PSS09]. 2. F´ary’s theorem: Every planar graph admits a crossing-free straight-line draw- ing [F´ar48, Tut60, Ste22]. Moreover, every 3-connected planar graph and its dual have simultaneous straight-line drawings in the plane such that only dual pairs of edges cross and every such pair is perpendicular [BS93]. 3. Koebe’s theorem: The vertices of every planar graph can be represented by nonoverlapping disks in the plane such that two of them are tangent to each other if and only if the corresponding two vertices are adjacent [Koe36, Thu78]. This immediately implies F´ary’s theorem. Preliminary version (July 26, 2017). To appear in the Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, and C. D. Tóth (editors), 3rd edition, CRC Press, Boca Raton, FL, 2017. Chapter 10: Geometric graph theory 259 4. Pach-T´oth theorem: Any graph that can be drawn in the plane so that its edges are represented by x-monotone curves with the property that any two of them either share an endpoint or properly cross an even number of times admits a crossing-free straight-line drawing, in which the x-coordinates of the vertices remain the same [PT04]. Fulek et al. [FPSS13]ˇ generalized this result in two directions: it is sufficient to assume that (a) any two edges that do not share an endpoint cross an even number of times, and that (b) the projection of every edge to the x-axis lies between the projections of its endpoints. 5. Grid drawings of planar graphs: Every planar graph of n vertices admits a straight-line drawing such that the vertices are represented by points belong- ing to an (n 1) (n 1) grid [FPP90, Sch90]. Furthermore, such a draw- ing can be found− × in O−(n) time. For other small-area grid drawings, consult [DF13]. 6. Straight-line drawings of outerplanar graphs: For any outerplanar graph H with n vertices and for any set P of n points in the plane in general position, there is a crossing-free geometric graph G with V (G)= P , whose underlying graph is isomorphic to H [GMPP91]. For any rooted tree T and for any set P of V (T ) points in the plane in general position with a specified element p P| , there| is a crossing-free straight-line drawing of T such that every vertex∈ of T is represented by an element of P and the root is represented by p [IPTT94]. This theorem generalizes to any pair of rooted trees, T1 and T2: for any set P of n = V (T ) + V (T ) points in general position in the plane, | 1 | | 2 | there is a crossing-free mapping of T1 T2 that takes the roots to arbitrarily prespecified elements of P . Such a mapping∪ can be found in O(n2 log n) time [KK00]. The analogous statement for triples of trees is false. 7. Alternating paths: Given n red points and n blue points in general position in the plane, separated by a straight line, they always admit a noncrossing alternating Hamiltonian path [KK03]. TURAN-TYPE´ PROBLEMS By Euler’s Polyhedral Formula, if a geometric graph G with n 3 vertices has no 2 crossing edges, it cannot have more than 3n 6 edges. It≥ was shown in [AAP+97] and [Ack09] that under the weaker condition− that no 3 (resp. 4) edges are pairwise crossing, the number of edges of G is still O(n). It is not known whether this statement remains true even if we assume only that no 5 edges are pairwise crossing. As for the analogous problem when the forbidden configuration consists of k pairwise disjoint edges, the answer is linear for every k [PT94]. In particular, for k = 2, the number of edges of G cannot exceed the number of vertices [HP34]. The best lower and upper bounds known for the number of edges of a geometric graph with n vertices, containing no forbidden geometric subgraph of a certain type, are summarized in Table 10.1.1. The letter k always stands for a fixed positive integer parameter and n tends to infinity. Wherever k does not appear in the asymptotic bounds, it is hidden in the constants involved in the O- and Ω-notations. Better results are known for convex geometric graphs, i.e., when the vertices are in convex position. The relevant bounds are listed in Table 10.1.2. For any convex geometric graph G, let χc(G) denote its cyclic chromatic number. Furthermore, let Preliminary version (July 26, 2017).
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